Thus, in order to make a prediction, a new linear predictor \(\langle \widehat{\omega }_{\lambda , n}(X), \cdot \rangle \)  is computed for every point \(X\)  and in particular, the VAW forecaster is non-linear. For background on the VAW forecaster and regret bounds we refer to [1]}, [2]}, [3]}. Below, for any predictor \(f(\cdot )\)  we denote \(R(f(X)) = \operatorname{\mathbb {E}}(Y - f(X))^2\) .
Our key observation is that the sequence of weights \(\widehat{\omega }_{\lambda ^*, j}(X)\)  for \(j = 1, \ldots , n\)  and \(\lambda ^* = \frac{dm^2}{b}\)  can be immediately translated into a non-linear estimator \(\widetilde{f}^{\operatorname{VAW}}_{\lambda ^*}(\cdot )\)  satisfying in the notation of the lower bound (REF ):
\(\operatorname{\mathbb {E}}R(\widetilde{f}^{\operatorname{VAW}}_{\lambda ^*}(X)) - R(\omega ^*_b) \lesssim \frac{dm^{2}}{n}\log \left(1 + \frac{r^{2}b^{2}n}{d^2m^2}\right).\) 
